Table Of ContentEffect of disorder on the conductance of a Cu atomic point contact
D. J. Bakker,∗ Y. Noat,† A. I. Yanson,‡ and J. M. van Ruitenbeek
Kamerlingh Onnes Laboratorium, Universiteit Leiden,
Postbus 9504, 2300 RA Leiden, The Netherlands
(Dated: February 1, 2008)
Wepresentasystematicstudyoftheeffectofthedisorderincopperpointcontacts. Weshowthat
peaksintheconductancehistogramofcopperpointcontactsshiftuponadditionofnickelimpurities.
The shift increases initially linearly with thenickel concentration, thusconfirming that it is dueto
disorder in the nanowire, in accordance with predictions. In general, this shift is modelled as a
resistance Rs whichis placed inseries with thecontact resistance Rc. However,weobtain different
Rs valuesfor thetwo peaks in thehistogram, Rs being larger for the peak at higher conductance.
2
0 INTRODUCTION individualatomsofthemetalunderinvestigation. Itwas
0 shown[9] thatthe conductance ofa one-atompoint con-
2 Quantum point contacts (QPC’s) are constrictions in tactisgovernedbythenatureofitsvalenceorbitals. For
n a conducting material whose size is of the order of the monovalent s-metals such as Na, Cu or Au, a one-atom
a Fermi wavelength of the electrons. The first quantum contact has a conductance of nearly 1 G0. On the other
J
point contacts were fabricated in a 2-dimensional elec- hand,transitionmetalssuchasNi, PdorPt,whichhave
7
tron gas (2DEG) in a semiconductor heterostructure d-valence orbitals, have a one-atom conductance of the
1 [1, 2]. Their conductance exhibits a series of steps as order of 1.6 G0.
v a function of constriction size. The plateaus in be- In experiments on both semiconductor QPC’s [1] and
8 tween these steps are positioned at integer multiples QPC’smadeofmonovalents-metalsithasbeenobserved
7 of G0 =2e2/h ≈ (12.9 kΩ)−1, the quantum of conduc- thatthequantizationoftheconductanceisneverperfect:
0
tance. This phenomenon is explained as a consequence the peaks in the histograms are shifted below quantized
1
0 of the quantization of the electronic energy levels, re- valuesmG0 (m =1,2,3...). Inmetallicpointcontacts,
2 sulting from the confinement of the electrons in the di- such a shift has been observed in gold [10, 11], sodium
0 rection perpendicular to the current. Signs of quantized [12], silver [11] and copper [11]. Two possible origins of
t/ behavior were also seen in the conductance of quantum theshifthavebeendiscussedintheliterature. Ithasbeen
a point contacts made from metals [3, 4, 5]. When a thin attributed to disorder either in the leads to the contact
m
metallic wireis slowlybroken,its conductanceas afunc- [1], or in the contact itself [13, 14, 16]. In all cases, a
d- tion of contact size shows smooth plateaus, alternated modelwasadoptedinwhichthe shiftisaccountedforby
n by steps of the order of G0. In this case, however, the an ‘extra’ resistance Rs that is placed in series with the
o plateaus are not necessarily horizontal or positioned at true contact resistance Rc. Gc ≡ 1/Rc is assumed to be
c integer multiples of G0. Moreover, when the two wire an exact integer multiple of G0, and it is then straight-
:
v ends are pressed together and broken again, the precise forward to determine Rs from the histograms. Typical
i shape of the trace differs from trace to trace. In order experimental values are 400 Ω for a 2DEG [1], 100 Ω for
X
to obtain fully reproducible results, a histogram is often sodium [12], 100-500 Ω for gold [10, 11], 500 Ω for silver
r
a created by projecting each conductance trace onto the [11] and 700 Ω for copper [11].
conductance axis and adding the contribution of several InalltheoreticalstudiesintodisorderinQPC’s,adding
thousands of traces. For many metals, the histograms disorder resulted in a downshift and smearing of the
show well-defined peaks which represent preferred con- peaks in the conductance histogram. In the work of
ductance values. Garc´ıa-Mochales et al. [15], for example, the nanowire
The physical origin of these peaks has been debated was simulated by a tight binding model. The disorder
heavily [6], especially the question whether they are in- is introducedbyrandomon-siteenergies,while the over-
deedrelatedto the quantizationofelectronic energylev- lap between atomic orbitals is described by a hopping
els or rather to the discrete and abrupt changes in the term between sites. The conductance was calculated as
cross-sectional area of the nanowires, which are caused a function of the Fermi energy, while the geometry was
by the fact that the contact must change its size by at keptconstant. Ahistogramwasthenconstructed,which
least one, discrete, atom. It was shown both experimen- showed peaks that were indeed shifted. The series re-
tally, by force measurements [7], and theoretically, by sistance (calculated by fitting the best resistance value
numerical simulations [8], that steps are correlated with centering simultaneously all peak maxima at the corre-
atomic rearrangementsof the contact. Thus, plateaus in sponding integer values of G0) were found to range from
the conductance traces correspond to stable atomic con- about10Ωforsmalldisorderto1000Ωforlargedisorder.
figurations. Apartfromits geometry,theconductanceof Bu¨rkiet al. [16] used a 2D-free electronmodel, with a
suchaconfigurationisdeterminedbytheconductanceof potential term in the Hamiltonian describing randomly
2
distributed impurities. The neck of the constriction in
the nanowire was elongated, while keeping the Fermi
energy constant. The histogram exhibited downshifted 2.0
peaksthatcouldbepositionedatintegermultiplesofG0 Pure Cu
by subtracting a series resistance of 525 Ω. 1.8 Cu99% Ni1%
Thus, while several authors have studied disorder in Cu90% Ni10%
QPC’s from a theoretical point of view, very little ex- Cu75% Ni25%
1.6
perimental study was devoted to it. Ludoph et al. [17] Cu65% Ni35%
studied conductance fluctuations as a function of bias Cu55% Ni45%
voltage in atomic-size contacts for pure metals. A con- ) 1.4 Cu50% Ni50%
.
u
sistent explanation was obtained in terms of scattering . Pure Ni
a
of the electrons on defects near the contact and inter- s ( 1.2
ference of various electron paths with the reflected wave nt
at the contact itself. The values obtained for the elastic u
o
scattering length, le, were used to estimate an effective c 1.0
f
series resistance that would result from this scattering, o
r
andthisresistancewasfoundtobeinagreementwiththe e 0.8
b
observedshiftsintheconductancepeaksnear1and3G0 m
for Cu, Ag and Au. On the other hand, the shift of the u
second peak was much larger and this led Ludoph et al. N 0.6
toproposethattheoriginofthispeakmaybeapreferred
atomic configuration,ratherthan conductance quantiza- 0.4
tion. Li et al. [18] studied the influence of the presence
of adsorbate molecules such as nitrogen or ethanol on
0.2
the conductance of a gold nanowire. Plateaus are still
observed in individual conductance traces, but their po-
sitions scatter more, resulting in a smearing of peaks in 0.0
0 1 2 3 4 5
the histogram,which depends on the strength of the ad-
Conductance (G )
sorbate used. This effect is interpreted as a result of 0
the scattering of conduction electrons by the adsorbate
molecules. It is noted that the first peak is less affected FIG. 1: Averaged conductance histograms at 10 mV bias.
than the others. Fromtoptobottom: purecopper,copperalloyscontaining1,
The work presented here is an experimental study of 20,25,35,45and50%nickel,andpurenickel. Allhistograms
have been normalized to the area below the curve and were
the effect on conductance histograms of various levels of
shifted along they-axisfor clarity.
disorder. For this purpose, we have measured the his-
togramsofcopper-nickelalloysforvariousnickelconcen-
trations, and compared them to the histogram of pure
copper. The disorder in the point contact is increases and 50 atomic %. The required amounts of copper and
with the nickel concentration. nickelweremeltedat1500◦C.Thealloyswerepulledinto
wireswithadiameterofabout200µm,whichweresubse-
quentlyannealedfor24hoursat900◦C.Metalliccontacts
EXPERIMENTAL METHOD
of atomic size were then obtained by using the so called
MechanicallyControllableBreakJunction(MCBJ)tech-
Our samples are wires made of random alloys of cop- nique. Apieceofwireisnotchedinthemiddleandglued
per and nickel. Two reasons have motivated the choice on a bendable phosphor-bronze substrate by two drops
for this alloy: Firstly, its phase diagram is particularly of epoxy. The sample is insulated from the substrate
simple,copperandnickelbeingcompletelymiscibleinall by a layer of kapton foil, and four measurement wires
concentrations. Secondly, the histograms of copper and are connected to it with a conducting glue. The sample
nickel are very different, as will be shown below. This is first cooled down to 4.2 K and then broken in vac-
makes it easy to recognize whether increasing the nickel uum (10−5 mbar at room temperature) by bending the
concentrationchangesthehistograminaqualitativeway, substrate. The bending is achieved by pressing a piezo
which would make it impossible to see the nickel atoms element against the substrate by means of a differential
as impurities in the copper lattice, or whether it induces screw. The high vacuum provided by the low tempera-
onlyquantitativechanges,e.g. ashiftofthe peaksinthe tureensuresatomicallycleancontactsafterrupture. The
copper histogram. two electrodes are then brought back into contact and
The nickel concentrations used are 1, 10, 25, 35, 45 their separationis controlledusing the piezo element. In
3
this way, the displacement between the two electrodes
canbe controlledwith a precisionofthe orderof 0.01˚A.
100 0.28
Thewireisbrokenandbroughtbackintocontactrepeat-
edlybyapplyingasawtoothvoltagetothepiezoelement. Bias 10 mV 0.26
Dspssauarummrorexippinrmllgeeepsatartwhoreeeledyrubme3crie0embaa0iesk0lauiitnsirynueg,ddrfsei.ovtdrai,dAgewuenasahclhihcluceisaorttlrnolheodgneyurtaccaaomttannnltdieacsaecvstbottulrwttiaawlacdgsoeesrf.droeoivgTffmeueorlraateerhnpnlyet-- ght of first peak (a.u.) 468000 BBiiaass 5100 0m mVV dth of first peak (G)000000.....1122268024
‘renewed’bypressingtheelectrodesdeepintoeachother. Hei Wi 0.14 Bias 10 mV
0.12 Bias 50 mV
20 0.10 Bias 100 mV
RESULTS
0.08
0 10 20 30 40 50 0 10 20 30 40 50
Ni concentration (%) Ni concentration (%)
In Fig. 1, the histograms of the pure elements Cu
and Ni and those of the copper alloys containing 1, 10,
FIG. 2: Dependence on Ni concentration of the height and
25,35,45 and50%nickelarepresented. These measure-
width of thefirst peak in thehistogram for several biases.
mentsweredoneusingabiasvoltageof10mV.Topmost,
the histogramof pure copper is shown. It is obtained by
averagingten histogramsmade on four different samples
of pure copper. Three peaks can be distinguished, posi- o 2.0
tionedatconductancevaluesG=1.0G0,1.7G0 and2.7 ve t m) 1.8 First peak
G0. Thefirstpeakhasahigheramplitudeandissharper ati h 1.6 Third peak
than the others. Close inspection makes clear that this el O 1.4
first peak is actually slightly below 1 G0. At the bottom e r (k 1.2
of Fig. 1, the histogram of pure nickel is shown. It is nc er 1.0
a p
vberroyaddififfresrtenpteafkrocmenttheeredcoaprpoeurnhdis1t.o6grGa0m,:anitdeaxhsiebciotnsda, sist cop 00..68
even broader, structure centered around 3.1 G0. s re ure 0.4
Looking at the histograms of the alloys, it is remark- e p 0.2
able that for all nickel concentrations the histograms re- eri 0.0
S
semble the histogram of pure copper; in particular, they -0.2
have a first peak a little below 1G0. Indeed, for biases 0 10 20 30 40 50
lower than 200 mV and for nickel concentrations up to Nickel concentration in alloy (%)
50%,thereisnosignofa”nickel”peakat1.6G0. Wegive
apossibleinterpretationofthiseffectinthenextsection.
FIG. 3: Series resistance corresponding to the shift relative
The second peak disappears rapidly when one increases
to pure copper of thefirst and third peaks in thehistograms
the nickelconcentration,due to shifting andbroadening.
for different nickel concentrations, at a bias of 10 mV. Axes
With increasing nickel concentrations, three things can are indicated for clarity.
be seen to happen to the other peaks in the histogram:
broadening, decreasing in amplitude and downshifting.
The width and height of the first peaks in the his- ductance values of the first and third peaks in the his-
togram are plotted in the right and left panel of Fig. 2, togram. In Fig. 3, the series resistance corresponding
respectively. As can be seen, the amplitude of the peak to this shift is plotted as a function of the percentage of
decreasesmonotonouslywithNiconcentration. Thiscan nickel in the alloy. Note that we do not calculate the
be explained as the result of increasing disorder. The shift with respectto integermultiples ofG0,but relative
disorderleadstoslightvariationsinthelast-plateaucon- to the measured positions of the corresponding peaks in
ductance of different scans. These variations cause the the histogram of pure Cu. Results again reproduce for
differentplateaustofallintodifferentbinsontheconduc- other biases. The shift increases roughly linearly with
tanceaxis,thuscausinglesshighpeaksinthehistogram. nickel concentration, up to 35% nickel. Two surprising
The sameeffect givesriseto the broadeningofthe peaks featuresofFig. 3mustbe noted. Firstly,the seriesresis-
that is observedfor Ni concentrationsup to 35%. Above tance is not the same for all the peaks in the histogram.
this concentration,the width decreasesto a minimum at Instead, the resistance corresponding to the shift of the
45%, after which it increases again. Fig. 2 also shows third peak is roughly two or three times higher than the
that the results reproduce for different biases. The main corresponding values for the first peak. The same con-
feature observedin Fig. 1 is the shift towardslowercon- clusion can be reached when considering the resistance
4
necessary to bring the first and third peaks of the pure atomspresentsegregateto the surfaceofthe cluster. All
copper histogram to the corresponding integer multiples the atomsformingthe contactinaMCBJcanbe viewed
of G0, namely, 60 ± 20 Ω and 460 ± 60Ω respectively. as being at the ‘surface’, to the extent that their coordi-
This is in agreement with earlier findings [11], but con- nationnumberislessthaninthebulk. Thustheremight
tradictorytothewidespreadpracticetobringallconduc- be a preference for Cu atoms to form the contact, which
tance peaks to integer multiples of G0 by subtracting a wouldleadtotheabsenceofNipeaksinthehistogramof
constant series resistance. The second surprising feature the alloy. However, this surface segregation takes places
of Fig. 3 is the decrease of the shift for Ni concentra- only at elevated (> 600 K) temperatures, while we per-
tions higher than 35%. This decrease correspondsto the formed our measurements at 4 K and at low (10 mV)
decrease in width that was already noticed in Fig. 2. bias, such that local heating of the contact is probably
Again the minimum is at 45% Ni. We discuss a possible not sufficient for segregation to happen [21]. Moreover,
interpretation of this effect in the next section. preliminary studies by molecular dynamics simulations
[22] don’t show such a segregation between Cu and Ni
atoms in the contact. Possibly, however, the repeated
DISCUSSION contact indentation and breaking provides enough mo-
bilityfortheatomstoobtainsomedegreeofsegregation.
The main difference between the histograms of cop- Another possibility to explain the similarity between
per and nickel, as shown in Fig. 1, is the position of the histograms of the alloys and that of pure copper is
the peaks. In particular, whereas copper shows a sharp assuming that the conductance of Cu-Ni constriction is
first peak nearly exactly at1 G0, nickelhas a broadfirst the same as that of a Cu-Cu constriction, namely, 1 G0.
peak centered around 1.6 G0. In both cases, however, For a contact of Cu50Ni50 in the form of two pyramids,
it seems reasonable to assume that the first peak in the this would lead to at least 75% of all last-atom contacts
histogram corresponds to the smallest possible contact: having a conductance of approximately 1 G0. Calcula-
only one atom bridging the gap between the electrodes. tions to verify this are under way.
This is supported by the fact that the conductance in- If the above picture is correct, adding nickel atoms
variably drops sharply to zero at the end of the plateau indeed doesn’t change the copper histogram in a quali-
at ∼ 1 (Cu) or ∼ 1.6 G0 (Ni). The difference in con- tative way. Instead, the nickel atoms only act as scat-
ductance between the different kind of atoms can be ex- terers for the electrons in the banks to the contact. One
plained by taking into account their valence orbitals. As would expect the amount of disorder in the banks to in-
already mentioned in the introduction, copper has one crease with the nickel concentration. In Fig. 3, it is
spherically symmetric s valence orbital, whereas the va- shown that the series resistance increases approximately
lence orbitals of nickel are d-type. It has been shown linearly with nickel concentration up to a nickel concen-
[9] that for monovalent s-type metals there is one, al- tration of 35%. From this we conclude that it is indeed
mostcompletelyopened, conductancechannelper atom, the disorder in the nanowire, in the form of impurities,
which givesrise to a 1-atomconductance of nearly 1 G0. thatcausesthedownshiftofthepeaksinthehistograms.
In contrast, the five valence orbitals of a d-metal give This is in accordance with the results of the calculations
rise to five partially opened channels, with a total con- cited above.
ductance in the range of 1.5 - 2.5 G0. However, the sudden decrease of the series resistance
Wenoticedthatthehistogramsforallthealloysresem- whengoingto45%nickel,accompaniedbyadecreaseof
ble the histogram of pure copper, in particular, the first the width of the first peak, suggest that the amount of
peak is below 1 G0, and the third is below 2.7 G0. How- disorder decreases when one increases the nickel concen-
ever,ifwe verycrudelymodelacontactofe.g. Cu50Ni50 tration to this value. In order to compare with the be-
as consisting of two pyramids touching eachother in the havior of bulk copper-nickel, we have measured the bulk
apexes, such that there are two constricting atoms, 25% resistivity of the alloys at room temperature as well as
of all contacts should end in a Cu-Cu constriction, 25% at T = 4.2 K. Results are presented in Fig. 4. It can be
shouldendinaNi-Niconstriction,and50%shouldhavea seenthat the low-temperatureresistivity initially closely
Cu-Niconstriction. Thisraisesthequestionwhywedon’t follows the RT resistivity and is linear with the nickel
see the influence of the nickel atoms in the histogram of concentration. Upon going to 45% nickel, however, the
this alloy, apart from the downshifting and smearing of low-temperature resistivity decreases, whereas the room
its peaks. Anexplanationfor this phenomenonmightbe temperature resistivity still increases,although less than
that the Ni atoms do not take part in the forming of the would be expected from a linear behavior. The decrease
contact. Thismightbeduetosurfacesegregation,which in the shift of the peaks in the copper-nickel histograms
is known to lead to a very strong copper enrichment of therefore probably reflects the decrease is resistivity of
thesurfacelayersinbulkCu-Ni[19],duetothelowersur- bulk copper-nickel. This shows that the origin of the se-
face energy of Cu atoms. Calculations for small Cu-Ni ries resistance is probably in the banks to the contact,
clusters [20] show the same phenomenon: all the copper and is not a (quantum mechanical) effect in the contact
5
ferred. It can be seen clearly that the series resistance
(cid:13) is not the same for all the peaks in the histogram. The
resistance necessary to shift the third peak in the his-
)(cid:13) 0.5(cid:13) 4.2 K(cid:13) tograms back to its value for pure copper is larger than
m
Room temperature(cid:13)
m 0.4(cid:13) forthe firstpeak. Thedifference increaseswithdisorder.
h This effect is not accounted for by any of the calcula-
O
o 0.3(cid:13) tions cited above. Possibly this difference in behavior
cr (cid:13) compared to the first peak can be accounted for by con-
mi m)(cid:13)100(cid:13) (cid:13) sidering the fluctuations in Cu/Ni ratio for contacts of
sistivity ( 00..12(cid:13)(cid:13) n free path (n 10(cid:13) T = 4.2 K(cid:13) (cid:13) taahrpeisaa.rstiizael,soergrbeygaationnono-fuNniifoartmomNsiawcoanycfernotmrattihoencdounetatcot
Re 0.0(cid:13) Mea 1(cid:13) 0(cid:13) 5(cid:13)10(cid:13)15(cid:13)20(cid:13)25(cid:13)30(cid:13)35(cid:13)40(cid:13)45(cid:13)50(cid:13)
Nickel concentration (%)(cid:13)
0(cid:13) 10(cid:13) 20(cid:13) 30(cid:13) 40(cid:13) 50(cid:13)
Nickel concentration (%)(cid:13)
CONCLUSION
FIG. 4: Room temperature and low temperature (4.2 K) re-
In conclusion, we have shown, firstly, that adding
sistivity of the copper-nickel alloys as function of nickel con-
nickel impurities to copper leads to an increase of the
centration. Inset shows the bulk mean free path, calculated
shift and smearingof the conductance peaks. This is ev-
from a Drudemodel.
idence that disorder in the nanowires is a main source
for the series resistance, in agreement with several theo-
itself. A likely explanation for the decrease is the onset retical papers. Up to a nickel concentration of 35%, this
of ferromagnetism in the nanowire. The ferromagnetic shiftincreasesroughlylinearlywiththenickelconcentra-
ordering will decrease the amount of spin-spin scatter- tion. Secondly, when the nickel percentage is increased
ing events by decreasing the magnetic fluctuations, even further, the shift decreases. The latter effect may be
though the nickel concentration increases. As a matter due to a ferromagnetic transition leading to a decrease
offact,bulkcopper-nickelbecomesmagneticat4.2Kfor of the resistance of the mesoscopic banks. It was shown
a nickel concentration of approximately 45% [24]. From that the bulk resistivityof Cu-Ni alloysfollows the same
Fig. 3itcanbeseenthattheresistancetendstoincrease behavior as the series resistance does, which is evidence
againwhen the nickelconcentrationis increasedto 50%, that the series resistance finds its origin in the banks to
which may be explained by assuming that once the al- the contactratherthan in the contactitself. Thirdly, we
loy is ferromagnetic, the amount of disorder in it again have shown that one single series resistance is not suffi-
increases with increasing nickelconcentration. The inset cient to shift all the peaks in the histogram back to the
in Fig. 4 shows the bulk mean free paths for the elec- respective quantizedvalues. Instead, it is found that the
trons in several of the alloys, calculated from the resis- series resistance needed for the third peak is larger than
tivities using the Drude model. An important difference that for the first. Fourthly, our measurements show that
between the bulk material and the quantum point con- adding nickel impurities has remarkably little influence
tacts is the behavior for low nickel concentration. It is on the global shape of the copper histogram, even if the
well known that the resistivity of metals at low temper- nickel concentration is raised to 50%. To explain this,
atures depends sensitively on the purity of the sample. we suggest that a Cu-Ni constriction has approximately
Thus,the meanfreepathinpurecopperat4.2Kwillbe the same conductance as a Cu-Cu constriction, or that
much larger than that in Cu99Ni1. On the other hand, there is a partial segregation of Ni atoms away from the
the histograms of these two materials are nearly indis- contact.
tinguishable. This means that in point contacts of both
materials, the mean free path is restricted by scattering
from the surface and lattice defects related to the geom-
etryofthecontact,ratherthanbyimpurities. Whenthe
ACKNOWLEDGEMENTS
concentrationofimpuritiesisraisedto10%,theystartto
have an appreciable effect. Thus, the mean free path in
thecontactregionofaQPCofpurecoppermustbeinbe- This research has been supported by a Marie Curie
tween those in bulk Cu99Ni1 and Cu90Ni10, i.e. between fellowship of the European Community under contract
5 and 30 nm. This is consistent with results obtained number HPMF-CT-1999-00196. We are grateful to I.K.
earlier for quantum point contacts of copper [17]. Yanson for fruitful discussions and to R.W.A. Hendrikx
From Fig. 3 a second important feature can be in- for providing the sample materials.
6
bacher, Phys. Rev.B, 56, 2208, (1997)
[12] J.M. Kranset al. Nature 375, 767 (1995)
[13] P. Garc´ıa-Mochales and P.A. Serena, Phys. Rev. Lett.
∗ Present address: Department of Applied Physics and 79, 2316, (1997)
DIMES, Delft University of Technology, Lorentzweg 1, [14] M.Brandbyge,K.W.JacobsenandJ.K.Nørskov,Phys.
2628 CJ Delft, The Netherlands Rev. B, 55, 2637, (1997)
† Presentaddress: GroupedePhysiquesdesSolides,Cam- [15] P. Garc´ıa-Mochales and P.A. Serena, Phys. Rev. Lett.
pus Jussieu tour 23, 2 Place Jussieu, 75251 Paris cedex 79, 2316 (1997)
05, France [16] J.Bu¨rki,C.A.Stafford,X.ZotosandD.Baeriswyl,Phys.
‡ Present address: Dept. of Physics, 510 Clark Hall, Cor- Rev. B, 60, 5000 (2000)
nell University,Ithaca, NY 14853 [17] B. Ludoph and J.M. van Ruitenbeek, Phys. Rev. B 61,
[1] B. J. van Wees et al.,Phys. Rev.Lett. 60, 848 (1988) 2273 (2000)
[2] D.A. Wharam et al.,J. Phys.C. 21 L887 (1988) [18] C. Z. Li, H. Sha and N. J. Tao, Phys. Rev. B 58, 6775
[3] J.M. Kranset al.,Phys.Rev. B 48, 14721 (1993) (1998)
[4] J.I. Pascual et al., Phys.Rev.Lett. 71, 1852 (1993) [19] See e.g. T. Sakurai et al.,Phys. Rev.B 34, 8379 (1986)
[5] L. Olesen et al.,Phys. Rev.Lett.72, 2251 (1994) [20] D.Tom´anek,S.MukherjeeandK.H.Bennemann,Phys.
[6] See e.g. A. I. Yanson and J. M. van Ruitenbeek, Phys. Rev. B 28, 665 (1983)
Rev.Lett. 79, 2157, (1997) and refs. therein. [21] For an estimate of the dependency of the local contact
[7] G. Rubio, N. Agra¨ıt and S. Vieira, Phys. Rev. Lett. 76, temperature on bias voltage, see H. E. van den Brom,
2302 (1996) PhD. thesis Leiden University,The Netherlands (2000)
[8] T.N.TodorovandA.P.Sutton,Phys.Rev.Letters, 70, [22] J. W. T. Heemskerk et al.,to be publised.
2138 (1993) [23] Y. Noat, N. D. Lang, M. van Hemert and J. M. van
[9] ElkeScheer et al.,Nature394 154, (1998) Ruitenbeek,to be published.
[10] J. L. Costa-Kr¨amer and N. Garc´ıa, Phys. Rev. B, 55, [24] See e.g. Constitution of binary alloys, ed. M. Hansen,
12910, (1997) McGraw-Hill book company,1953
[11] K. Hansen, E. Lægsgaard, I. Stensgaard and F. Besen-
